To find the inverse of a relation, we swap each ordered pair's $x$-coordinate and $y$-coordinate. The inverse of the relation $R = \{(1, 0), (3, -8), (4, -3), (7, -5), (9, -1)\}$ is obtained by flipping the pairs.

Here’s the process:

• The inverse of $(1, 0)$ becomes $(0, 1)$
• The inverse of $(3, -8)$ becomes $(-8, 3)$
• The inverse of $(4, -3)$ becomes $(-3, 4)$
• The inverse of $(7, -5)$ becomes $(-5, 7)$
• The inverse of $(9, -1)$ becomes $(-1, 9)$

Thus, the inverse of the given relation is: $R^{-1} = \{(0, 1), (-8, 3), (-3, 4), (-5, 7), (-1, 9)\}$

Would you like further explanation or have any questions?

Here are 5 related questions to extend your understanding:

1. What does it mean for a relation to have an inverse?
2. How do you determine if a relation is a function?
3. Is the inverse of a function always a function? Why or why not?
4. Can you graph both the relation and its inverse to visualize the process?
5. How do you determine if an inverse relation is also a function?

Tip: For a function to have an inverse function, it must be both one-to-one (injective) and onto (surjective).